Working with 1 asparagus spear at a time, use a vegetable peeler to shave spears into long, thin shavings. Transfer to a medium bowl (the tips will snap off as spears get thinner; add to bowl). Combine grated Parmesan and lemon juice in a small bowl and slowly whisk in oil until well blended. Season vinaigrette generously with salt and pepper. Drizzle vinaigrette over shaved asparagus and toss to coat. Divide asparagus salad among plates. Use peeler to shave more Parmesan over salad.